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#Python 3, 196 170 160 bytes

Awkwardly long, still golfing:

y=""
for C in input():
 for a in "[](){}":
  if C==a:y+=C
 y=y.replace("()","")
x=y
for r in y:x=x.replace("{}","")
for s in y:x=x.replace("[]","")
print(x=="")